Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:54:39Z http://mathoverflow.net/feeds/question/107414 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107414/matroid-representable-over-mathbbr-but-not-over-mathbbq Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$? Jeremy Martin 2012-09-17T18:46:47Z 2012-09-17T19:08:50Z <p>Does there exist a matroid that is representable over $\mathbb{R}$ but not over $\mathbb{Q}$?</p> <p>In particular, can one give a positive answer using a nonrational polytope, i.e., a combinatorial polytope that cannot be realized as the convex hull of rational vertices? (Such things do exist; see, e.g., p.94 of Grünbaum's <em>Convex Polytopes</em>.) The vertex sets of the faces of a convex polytope certainly form the flats of a matroid, but it's not clear to me why the same matroid could not be realized by affine dependences of a set of points not in convex position.</p> http://mathoverflow.net/questions/107414/matroid-representable-over-mathbbr-but-not-over-mathbbq/107416#107416 Answer by Igor Pak for Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$? Igor Pak 2012-09-17T19:07:33Z 2012-09-17T19:07:33Z <p>Jeremy, on the very same page 94 you will find a "point and line configuration" called <em>Perles configuration</em> which when viewed as set ov vectors in $\Bbb R^3$ is a matroid that is realizable over $\Bbb Q[\sqrt{5}]$ but not over $\Bbb Q$. In <a href="http://www.math.ucla.edu/~pak/book.htm" rel="nofollow">my book</a> I even prove it (Ex 12.3) - sorry to make a plug, this is the only place with a proof I know. </p>